- \( \left.\begin{array}{r}5 u+3 T=5 \\ 4 u+7 T=27\end{array}\right\} \)

To solve the system of equations \(5u + 3T = 5\) and \(4u + 7T = 27\), you can use the substitution or elimination method. A quick way is to multiply the first equation by 4 and the second by 5 to align the \(u\) terms. This gives you: 1. \(20u + 12T = 20\) (Equation 1 multiplied by 4) 2. \(20u + 35T = 135\) (Equation 2 multiplied by 5) Now, subtract the first modified equation from the second: \[ (20u + 35T) - (20u + 12T) = 135 - 20 \] This simplifies to: \[ 23T = 115 \implies T = 5 \] Substituting \(T = 5\) back into the first equation: \[ 5u + 3(5) = 5 \implies 5u + 15 = 5 \implies 5u = -10 \implies u = -2 \] Thus, the solution to the system is \(u = -2\) and \(T = 5\). Now, for some additional info: Understanding the historical context of systems of equations can be quite fascinating. The concept dates back to ancient civilizations, such as the Babylonians and Egyptians, who used geometric methods to solve problems that we would now express as linear equations. These early mathematicians laid the groundwork for algebra, which blossomed during the Islamic Golden Age with scholars like Al-Khwarizmi, earning him the title of “father of algebra.” In terms of real-world application, systems of equations are vital in numerous fields, from economics to engineering. For example, businesses use these equations to maximize profits by determining how many units of products to produce based on cost constraints and predicted demand. In engineering, they model relationships between different materials and forces, ensuring that structures are safe and efficient. This mathematical framework helps professionals make informed decisions that impact our daily lives!